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Econ-261 · economics · Economics

Digital revenue-contingent finance with imperfect observability

LLM status Open unverified

Statement

ECON-087 | R | Source-backed agenda; exact formalization is editorial

Definitions and formal target

In a two-date model, a lender advances $L$ at date zero. The borrower chooses investment, date-zero consumption and transaction diversion $d$. Date-one true resources $R$ and digital record $Z$ follow a specified technology/measurement law depending on these choices. Contract $t(Z)$ induces a selected best response. For utility $u_0(c_0)+\beta u_1(c_1)$, with $c_1=R-t(Z)-k(d)$, compare admissible contracts subject to \[ \mathbb E[t(Z)]\geq(1+r_f)L+C_{\rm proc},\qquad \mathbb E[U(t)]\geq U^{\rm outside}. \] Include a stated limited-liability/default rule and feasible consumption. Identify whether revenue contingency improves risk sharing relative to fixed repayment after record error and manipulation.

Clarifications and required assumptions

Define the date-zero budget linking $L$, investment and consumption; loan size is not an unexplained additional utility argument. Distinguish true revenue, cash available for repayment and recorded revenue. If diversion cost is a real expenditure, specify when it is paid; the formula uses date one. A rule $t(Z)\leq Z$ does not automatically imply feasibility against true resources when records are noisy. Contract-dependent revenue/diversion must be recomputed rather than treated as an exogenous sample. Funding principal and processing costs enter lender accounts once.

Variants and changes of scope

  • Perfect verifiability, noisy records and strategic diversion are different information models.
  • Revenue shares, profit shares and fixed loans define different repayment bases.
  • Dynamic repeated lending adds reputation and learning; its incentive conditions differ from the two-date example.

Audit conclusion and source relationship

[S1] and [S2] support digital/performance-contingent finance. The record-manipulation model is an editorial extension, not a precise contract conjecture stated by the microfinance review.

References and exact source roles

[S1] Jing Cai, Muhammad Meki, Simon Quinn, Erica Field, Cynthia Kinnan, Jonathan Morduch, Jonathan de Quidt, and Farah Said (2025). Microfinance. VoxDevLit 3(3), January 2025.

Verified locator / source role: Primary publisher page checked for review identity, authors, version and background. The specific published agenda is corroborated in [S2]; the exact formal target is editorial.

https://voxdev.org/voxdevlit/microfinance

[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.

Verified locator / source role: Topic heading: Microfinance. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-261-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-261

Revenue sharing with noisy records and costly diversion

Problem ID: D86-257. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and timing

Can revenue-contingent digital contracts improve borrower welfare while remaining financeable and incentive compatible when receipts can be hidden? The microfinance agenda [Vox] raises this institutional question. We solve an affine-contract model in which measurement noise and diversion jointly limit risk sharing. The affine restriction is substantive; no optimum over all measurable contracts is claimed.

At date zero a loan $L>0$ is invested in a fixed project. Its date-one true revenue is $R=\mu+\sigma_R\xi$, where $\sigma_R>0$ and $\xi$ takes values $\pm1$ with equal probability. A recorded receipt is \[ Z=R-d+\sigma_\eta\zeta, \qquad\sigma_\eta\ge0, \] where $\zeta$ is an independent symmetric sign. After contracting but before either shock, the borrower chooses diversion $d\in[0,1/\kappa]$, with real resource cost $\kappa d^2/2$, $\kappa>0$. Diverting changes the record but does not destroy gross project revenue beyond that cost. Date-zero consumption is fixed across contracts. The borrower has date-one CARA utility $-e^{-\rho c_1}$, $\rho>0$, and \[ t(Z)=a+bZ,\quad b\in[0,1],\qquad c_1=R-t(Z)-\frac\kappa2d^2. \] The lender's required expected repayment is $F=(1+r_f)L+C$, with funding rate $r_f$ and processing cost $C$. Principal and funding costs occur only in this requirement. All primitive parameters are known to both parties.

Assume the following sufficient, deliberately conservative bounds: \[ F\ge\sigma_R+\sigma_\eta+\frac1{4\kappa},\qquad \mu-F>\sigma_R+\sigma_\eta+\frac3{2\kappa}. \tag{1} \] They will guarantee nonnegative transfers and feasible consumption for every contract in the break-even family and every allowed deviation. The model does not require transfers to be below the possibly noisy recorded receipt; liability is evaluated against true available resources.

Incentives and financeability

Theorem (An incentive-compatible affine frontier): For any fixed $a,b$, the borrower's unique diversion choice is $d=b/\kappa$. In a borrower-optimal financeable affine contract the expected-repayment constraint binds, so \[ a=F-b\mu+\frac{b^2}{\kappa}. \tag{2} \] Under (1), every contract in (2) satisfies $t(Z)\ge0$ and $R-t(Z)-\kappa d^2/2>0$ for all $d\in[0,1/\kappa]$ and both shock values. At its incentive-compatible diversion, the borrower's certainty equivalent is \[ \operatorname{CE}(b)=\mu-F-\frac{b^2}{2\kappa} -\frac1\rho\log\cosh\big(\rho(1-b)\sigma_R\big) -\frac1\rho\log\cosh(\rho b\sigma_\eta). \tag{3} \]
Proof: Conditional on the chosen contract, consumption equals $\mu-a-b\mu+bd-\kappa d^2/2+(1-b)\sigma_R\xi-b\sigma_\eta\zeta$. The random part does not depend on $d$. Since utility is increasing, maximize the deterministic concave quadratic $bd-\kappa d^2/2$. Its unique maximizer is $b/\kappa$, within the allowed interval. Expected repayment is then $a+b\mu-b^2/\kappa$. Lowering $a$ increases consumption in every state without changing diversion; hence borrower optimality makes the repayment requirement bind, yielding (2).

For an arbitrary deviation, substitution gives \[ t=F+\frac{b^2}{\kappa}+b\sigma_R\xi+b\sigma_\eta\zeta-bd \ge F-\sigma_R-\sigma_\eta-\frac1{4\kappa}\ge0, \] using $b^2-b\ge-1/4$. Consumption is \[ \mu-F-\frac{b^2}{\kappa}+(1-b)\sigma_R\xi-b\sigma_\eta\zeta+bd-\frac\kappa2d^2 >0 \] by the second inequality of (1), since $b^2/\kappa\le1/\kappa$, $\kappa d^2/2\le1/(2\kappa)$, and $bd\ge0$. Thus no infeasible deviation was used in the incentive comparison. At $d=b/\kappa$, consumption simplifies to $\mu-F-b^2/(2\kappa)+(1-b)\sigma_R\xi-b\sigma_\eta\zeta$. Independence of the signs makes its exponential moment the product of two hyperbolic cosines; $-\rho^{-1}\log{\mathbb{E}} e^{-\rho c_1}$ is exactly (3). ∎

The optimal risk-sharing rate

Theorem (Unique sharing rate and comparative statics): The certainty equivalent (3) has a unique maximizer $b_*\in(0,1)$, characterized by \[ \frac{b_*}{\kappa} =\sigma_R\tanh\big(\rho(1-b_*)\sigma_R\big) -\sigma_\eta\tanh(\rho b_*\sigma_\eta). \tag{4} \] For comparisons where (1) remains satisfied, $b_*$ strictly increases with $\kappa$ and weakly decreases with measurement noise $\sigma_\eta$, strictly for a positive change from a positive noise level. Relative to fixed repayment $b=0$ with the same funding requirement, the optimum strictly improves borrower welfare.
Proof: Differentiate (3): \[ \operatorname{CE}'(b)=-b/\kappa+ \sigma_R\tanh(\rho(1-b)\sigma_R) -\sigma_\eta\tanh(\rho b\sigma_\eta), \] \[ \operatorname{CE}''(b)=-1/\kappa -\rho\sigma_R^2\operatorname{sech}^2(\rho(1-b)\sigma_R) -\rho\sigma_\eta^2\operatorname{sech}^2(\rho b\sigma_\eta)<0. \] The first derivative is positive at zero and negative at one. This proves existence, uniqueness, interiority, and (4). At an interior root, the derivative of $\operatorname{CE}'$ with respect to $\kappa$ is $b/\kappa^2>0$. The derivative with respect to $\sigma_\eta$ is $-\tanh(\rho b\sigma_\eta)-\rho b\sigma_\eta\operatorname{sech}^2(\rho b\sigma_\eta)$, nonpositive and strictly negative when noise is positive. Implicit differentiation, dividing by the negative second derivative, gives the stated signs. Finally $\operatorname{CE}'(0)>0$ implies a strict gain for some sufficiently small positive $b$, and hence at the maximizer. ∎
The diversion cost increases with sharing because the borrower saves $b$ units of repayment per unit of hidden receipt. The funding adjustment in (2) passes the resulting expected repayment shortfall back through the intercept. This is why the equilibrium welfare loss is $b^2/(2\kappa)$ despite repayment being perfectly forecast by the lender.

If the borrower's outside option has certainty equivalent $\overline c$ and the same date-zero consumption, the affine project is voluntarily financeable exactly when $\operatorname{CE}(b_*)\ge\overline c$. This follows because certainty equivalents preserve CARA utility rankings and $b_*$ maximizes them over all lender-feasible affine contracts. If the inequality fails, relaxing participation is not a solution: no contract in this family is implementable. Fixed repayment can fail participation while the sharing contract succeeds, for any outside equivalent strictly between their two values.

Remaining gap

The model gives a measurable tradeoff among repayment smoothing, record quality, and manipulation. It does not identify those primitives from transaction data, solve nonlinear contracting, permit post-revenue diversion choices, or endogenize project scale. CARA utility and bounded two-point shocks are explicit restrictions used to maintain feasible consumption and derive exact welfare. Digital records need not satisfy the independent-noise assumption in practice. Establishing welfare dominance under endogenous reporting technology and competition between lenders remains part of the catalogue problem.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives incentive-compatible affine repayments, feasible borrower consumption and a uniquely optimal revenue-sharing rate with explicit noise and diversion comparative statics. Optimal nonlinear contracts, endogenous investment and reporting technology, later diversion choices and empirically identified welfare comparisons remain outside the solved benchmark.

References

  • [Vox] J. Cai, M. Meki, S. Quinn, E. Field, C. Kinnan, J. Morduch, J. de Quidt and F. Said (2025), "Microfinance," VoxDevLit 3(3). Living literature review. This motivates the contract-design agenda; the affine model and calculations are the explicitly restricted analysis here.

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